Construction of Bivariate Copulas on a Multivariate Exponentially Weighted Moving Average Control Chart

The control chart is an important tool in multivariate statistical process control (MSPC), which for monitoring, control, and improvement of the process control. In this paper, we propose six types of copula combinations for use on a Multivariate Exponentially Weighted Moving Average (MEWMA) control chart. Observations from an exponential distribution with dependence measured with Kendall’s tau for moderate and strong positive and negative dependence (where ) among the observations were generated by using Monte Carlo simulations to measure the Average Run Length (ARL) as the performance metric and should be sufficiently large when the process is in-control on a MEWMA control chart. In this study, we develop an approach performance on the MEWMA control chart based on copula combinations by using the Monte Carlo simulations.The results show that the out-of-control (ARL1) values for were less than for in almost all cases. The performances of the Farlie-Gumbel-Morgenstern Ali-Mikhail-Haq copula combination was superior to the others for all shifts with strong positive dependence among the observations and . Moreover, when the magnitudes of the shift were very large, the performance metric values for observations with moderate and strong positive and negative dependence followed the same pattern.


Introduction
Multivariate Statistical Process Control (MSPC) is an important method for process monitoring, control and improvement in many areas such as engineering, economics, environmental statistics, finance and etc. For example, in automotive production quality control depends on correlated variables such as the lifetimes of the components in the engine, etc. A control chart is a common tool for MSPC for detecting changes in the vector means of the process. Multivariate control charts are generalizations of their univariate counterparts [1]. Hotelling's T 2 was the first multivariate control chart [2], followed by the Multivariate Exponentially Weighted Moving Average (MEWMA) control chart as a better alternative for detecting small shifts in the process vector mean [3,4]. Most multivariate detection procedures are based on the assumption that the observations are independent and identically distributed (i.i.d.) and follow a multivariate normal distribution. However, many processes are non-normal and correlated, so multivariate control charts need to be able to cope with related joint distributions. Hence, Kuvattana et al. [5] and Sukparungsee et al. [6] introduced the copula to address this requirement.
Copulas are functions that join multivariate distributions to their one-dimensional marginal distribution functions in which the one-dimensional margins are uniform on the interval (0,1) [7]. They are used to explain the dependence between random variables and are based on a representation of Sklar's theorem [8]. A new way of constructing asymmetric copulas was introduced by Mukherjee et al. [9], and later on copulas have been applied to MSPC [10]. Several other studies have proposed and compared the performance of bivariate copulas on the multivariate control charts [11][12][13][14]. Herein, we present the efficiency of the combinations of bivariate copulas constructed for shifts in the process vector mean on a MEWMA control chart when observations follow an exponential distribution.

Research Methodology
This paper is organized into the following sections: in section 2.1 the multivariate exponentially weighted moving average (MEWMA) control chart. Section 2.2, we review copulas function and constructing bivariate copulas. Section 2.3 describes the dependence measure of data and finally section 2.4 provides the ARL and the simulation study.

The Multivariate Exponentially Weighted Moving Average (MEWMA) Control Chart
The MEWMA control chart was first developed by Lowry et al. [4]. The given observations from a d-variate Gaussian distribution , for i = 1,2,. . . , can be defined as (1) where is a vector of variable values from the data and is a diagonal matrix with entries , for and .
The quantity plotted on the control chart is , where When on the interval (0,1) (as assumed in this study), the control chart signals a shift in the mean vector when where H is the control limit chosen for the desired in-control process. Generally, the Average Run Length (ARL) can be used to measure the performance of a MEWMA control chart. It depends on the degree of dependence between the variables measured using the covariance matrix and the scalar-weighted associated with the past observations. We consider a bivariate EWMA control chart and the control limit H for the in-control process ARL 0 = 370.

Copulas Function and Constructing Bivariate Copulas
Theoretically, for the copula function according to Sklar's theorem [8] for a bivariate case, let X and Y be continuous variables with joint distribution function G and marginal cumulative distributions and , respectively. Consequently, with copula where is a parameter of the copula. Theoretically, let A and B be bivariate copulas. It follows that , where is a copula with parameters and [15]. If , then C 1,1 = A, and if then u) we have an asymmetric copula.
In accordance with Khoudraji's device [16], let C be symmetric copula , where is independence copula. A family of asymmetric copulas with parameters , that includes C as a limiting case is given by . (3)

Dependence Measure of the Data
Generally, a copula can be used in the study of the dependence of association between random variables by Kendall's tau, which we implemented in this study (Table-1). Let X and Y be continuous random variables with copula C, then Kendall's tau is given by

The ARL and the Simulation Study
Theoretically, the ARL is an average number of points that must be plotted before the out-of-control condition occurs. ARL is classified into ARL 0 and ARL 1 . ARL 0 is the average number of observations before the first out-of-control point, while ARL 1 is the average number of observations when the process is out-of-control. The expectations of ARL 0 and ARL 1 can be respectively expressed as for (4) for (5) where is the change point time, is the stopping time, and is the expectation under the assumption that the change point occurs at We ran a Monte Carlo simulation using R statistical software [17][18][19][20] with the 50,000 rounds and a sample size of 6,000. The observations were generated from a copula based on an exponential distribution with mean = 1 (for the in-control process) and shifts at level 0.01, 0.05, 0.1, 0.5, 1, and 5 (for the out-of-control process). The performance of the MEWMA control chart was assessed for = 0.05 and 0.10. For all combinations of copulas, setting corresponds to Kendall's tau for moderate and strong positive and negative dependence ( = 0.5, -0.8).

Results
The simulation results are reported in Tables 2 to 9, in which the results are only empirical. The aim of the study was to optimize the parameters for constructing bivariate copulas ( ), as shown in Equation (3), for which we used the Maximum pseudo-likelihood estimator method [21]. For the in-control process on the MEWMA control chart, the desired ARL 0 = 370 was set for each copula combination. The results in Tables 2 and 3 Tables 8 to 9 show strong negative dependence ( ).        The results in Tables 2 to 9 show that the ARL 1 values  for were less than those for in almost all cases. The results in Tables 2 and 3 indicate that the Clayton Ali-Mikhail-Haq (AMH) copula combination was superior to the others in almost all cases. Meanwhile, with strong positive dependence ( ) and , Farlie-Gumbel-Morgenstern (FGM) AMH attained the minimum ARL 1 with all shifts (Table 4). Meanwhile, for moderate negative dependence ( ), Clayton FGM attained the minimum ARL 1 with shift values at 0.01 and 0.05 (Table 6). For the results for strong negative dependence ( ) and (Table 8), the performance of FGM AMH was superior to the others with shift values at 0.5 and 1. However, when the magnitude of the shift was large ( ), the performances of all of the copula combinations for moderate and strong positive and negative dependence were the same.

Conclusions
In this study, we investigated closed-form approximations of the ARL for MEWMA control charts using bivariate copulas constructed via Khoudraji's device, and we used Monte Carlo simulation when the marginal of the variables was exponential with . The simulation results suggest that there were no meaningful differences between the performances of the bivariate copulas at a very large shift ( ) when the observations had moderate and strong positive and negative dependence. In addition, the performances of the constructed bivariate copulas were superior to a single copula [5] for a moderate shift in a process on a MEWMA control chart. For further research, we could use the real data to compare the simulation results.